The graph og `g(x)=x^(3)+1` was translated 4 units to the right and 2 units up, resulting in a new graph h(x). What is the value of h(3.7) ?

Options

A. `0.973`
B. `1.784`
C. `1.973`
D. `2.973`

Explanation

To solve the problem step by step, we will follow the transformations of the function \( g(x) = x^3 + 1 \) and then evaluate \( h(3.7) \).

### Step 1: Understand the Original Function
The original function is given by:
\[ g(x) = x^3 + 1 \]

### Step 2: Translate the Function 4 Units to the Right
When a graph is translated 4 units to the right, we replace \( x \) with \( x - 4 \). Thus, the new function \( v(x) \) becomes:
\[ v(x) = (x - 4)^3 + 1 \]

### Step 3: Translate the Function 2 Units Up
Next, we translate the function \( v(x) \) 2 units up. This means we add 2 to the entire function:
\[ h(x) = v(x) + 2 = (x - 4)^3 + 1 + 2 \]
Simplifying this gives:
\[ h(x) = (x - 4)^3 + 3 \]

### Step 4: Evaluate \( h(3.7) \)
Now we need to find \( h(3.7) \):
\[ h(3.7) = (3.7 - 4)^3 + 3 \]
Calculating \( 3.7 - 4 \):
\[ 3.7 - 4 = -0.3 \]
Now we calculate \( (-0.3)^3 \):
\[ (-0.3)^3 = -0.027 \]
Adding 3 to this result:
\[ h(3.7) = -0.027 + 3 = 2.973 \]

### Final Answer
Thus, the value of \( h(3.7) \) is:
\[ \boxed{2.973} \]